Update notifications are disabled by default, you can enable them here:
As for egui.exe, it doesn't run unless you click the icon to open the GUI. You should see eguiproxy.exe among running processes if GUI is not active.
For an overview of devices that run ESET activated with your license key, sign up for ESET HOME (https://home.eset.com) and add your license key to the License manager where you can control it and deactivate it on particular devices.